Unit 10 Section 4 : Multiplication Law For Independent Events

If a particular outcome or event in one experiment does not affect the probabililty of outcomes or events in a second experiment, then the
outcomes or events are said to be independent.

For example, if you roll a dice twice, then the outcome in the first roll will not affect the probabilities of the outcomes when the dice is rolled a second time.
The results of the two dice rolls are independent of each other.

Multiplication Law
If X and Y are two independent events with probabilities P(X) and P(Y) respectively, then the probability that X and Y will both happen
is found by multiplying the two probabilities together:

P(X AND Y) = P(X) × P(Y)

With this we can calculate probabilities of combined outcomes when the original experiment outcomes are not equally-likely.

Example Questions

Example Question 1

Two fair dice are rolled. Use a probability tree diagram to determine the probability of obtaining:
(a) two sixes,
(b) no sixes,
(c) exactly one six.

We are only interested if the dice shows a six or not, so we have two outcomes, "six" and "not six".
The probability of "six" is and the probability of "not six" is .

We start by drawing a tree diagram to show the outcomes:

Then we write on the probability for each branch.
Each set of branches should add up to 1 (e.g. + = 1).

Finally, we work out the probability of each outcome by multiplying the probabilities along the branches which lead to that outcome.
For example, the probability P(six AND not six) is found by multiplying P(six) by P(not six).

Note how all the outcome probabilities also add up to 1 - this will always happen and is a useful thing to check.

So, the answers are as follows:

(a) P( two sixes ) = P( six , six ) =
(b) P( no sixes ) = P( not six , not six ) =

To find the probability of "exactly one six", we need to combine the outcomes for ( six , not six ) and ( not six , six ), as they both have "exactly one six".
We combine these outcomes by adding their probabilities:

(c) P( exactly one six ) = P ( six , not six ) + P( not six , six ) =

Example Question 2

Russell is playing in a cricket match and a game of football at the weekend.
The probability that his team will win the cricket match is 0.7, and the probability of winning is 0.9 in the football.
Assume that the results in the matches are independent - they do not affect each other.

What is the probability that his team will win in both matches?

Although we could draw a probability tree in this situation, we do not need to.
We can simply use the multiplication law:

While a probability tree is often useful, it is worth checking whether simpler questions can be answered without one.

Practice Questions

Work out the answer to each of these questions then click on the button marked
to see whether you are correct.

Practice Question 1

A bag contains 4 red balls and 3 green balls.
A ball is taken at random, and then put back. A second ball is then taken from the bag.
The results are independent because the first ball is put back before the second is taken out.

(a) Draw a probability tree, showing the possible outcomes and their probabilities.

(b) What is the probability that:

(i) both balls are the same colour

(ii) the balls are of different colours

(iii) at least one of the balls is green

Practice Question 2

On her way to work, Sylvia drives through three sets of traffic lights.
The probability of each set of lights being green is 0.3.

What is the probability that they are all green?

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Exercises

Work out the answers to the questions below and fill in the boxes. Click on the
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then will appear and you should move on to the next
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the answer.

Question 1
A bag contains 3 red balls and 2 blue balls. A ball is taken at random from the bag and then put back.
A second ball is then taken out of the bag. The probability tree for this situation is shown below.

What is the probability that:

(a) both balls are red?

(b) both balls are the same colour?

(c) at least one of the balls is red?

Question 2
A six-sided dice has the numbers 1, 1, 2, 3, 5 and 8 on its faces. The dice is rolled twice.

(a) Fill in the probabilities on the tree diagram below. Remember to cancel down any fractions.

(b) What is the probability of obtaining:

(i) two even numbers?

(ii) at least one even number?

(iii) no even numbers?

Question 3
A coin is biased so that the probability of heads P(H) = 0.4 and the probability of tails P(T) = 0.6.
The coin is flipped twice and the results are recorded.

(a) Fill in the probabilities on the tree diagram below. Give all your answers as decimals.

(b) What is the probability of obtaining:

(i) two heads?

(ii) exactly one head?

(iii) less than two heads?

Question 4
The spinner shown below is spun three times.

(a) Fill in the missing probabilities in the tree diagram. Remember to cancel any fractions down.

(b) What is the probability of obtaining:

(i) three reds?

(ii) exactly two reds?

(iii) at least one red?

You will need some paper and something to write with for the next few questions.
You should draw a tree diagram if necessary in each question to work out the probabilities.
You can then type in the answers to each question in the boxes and check them.

Question 5
The spinner shown in the diagram below is spun twice.

Use a tree diagram to determine the probability of obtaining:

(a) two reds?

(b) at least one red?

(c) no reds?

Question 6
A coin has been weighted, so that the probability of getting a head is 0.2 and the probability of getting a tail is 0.8.
The coin is thrown twice.

What is the probability of obtaining:

(a) 2 heads?

(b) no heads?

(c) at least one head?

Question 7
A bag contains 1 red ball, 2 green balls and 4 yellow balls. A ball is taken from the bag at random.
The ball is then put back, and a second ball is taken at random from the bag.

What is the probability that:
(a) both balls are the same colour?

(b) no yellow balls are taken out?

(c) at least one yellow ball is taken out?

Question 8
Each of 10 balls is marked with a different number from 1 to 10. One ball is taken at random and then replaced.
A second ball is then taken at random.

Determine the probability that:
(a) both balls taken are marked with the number 5?

(b) both balls taken have even numbers?

(c) both balls taken have numbers which are multiples of 3?

(d) at least one of the balls taken has a number greater than 2?

Question 9
On his way to work, Paul has to pass through 2 sets of traffic lights.
The probability that the first set of lights is green is 0.5, and the probability that the second set of lights is green is 0.4.

What is the probability that both sets of lights are green?

You have now completed Unit 10 Section 4

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